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Factor a. 2 x 2 + 9 x + 9 b. 6 x 2 + x 1

a. ( 2 x + 3 ) ( x + 3 ) b. ( 3 x −1 ) ( 2 x + 1 )

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Factoring a perfect square trinomial

A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the product of the two terms and the square of the last term.

a 2 + 2 a b + b 2 = ( a + b ) 2 and a 2 2 a b + b 2 = ( a b ) 2

We can use this equation to factor any perfect square trinomial.

Perfect square trinomials

A perfect square trinomial can be written as the square of a binomial:

a 2 + 2 a b + b 2 = ( a + b ) 2

Given a perfect square trinomial, factor it into the square of a binomial.

  1. Confirm that the first and last term are perfect squares.
  2. Confirm that the middle term is twice the product of a b .
  3. Write the factored form as ( a + b ) 2 .

Factoring a perfect square trinomial

Factor 25 x 2 + 20 x + 4.

Notice that 25 x 2 and 4 are perfect squares because 25 x 2 = ( 5 x ) 2 and 4 = 2 2 . Then check to see if the middle term is twice the product of 5 x and 2. The middle term is, indeed, twice the product: 2 ( 5 x ) ( 2 ) = 20 x . Therefore, the trinomial is a perfect square trinomial and can be written as ( 5 x + 2 ) 2 .

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Factor 49 x 2 14 x + 1.

( 7 x −1 ) 2

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Factoring a difference of squares

A difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when the two factors are multiplied.

a 2 b 2 = ( a + b ) ( a b )

We can use this equation to factor any differences of squares.

Differences of squares

A difference of squares can be rewritten as two factors containing the same terms but opposite signs.

a 2 b 2 = ( a + b ) ( a b )

Given a difference of squares, factor it into binomials.

  1. Confirm that the first and last term are perfect squares.
  2. Write the factored form as ( a + b ) ( a b ) .

Factoring a difference of squares

Factor 9 x 2 25.

Notice that 9 x 2 and 25 are perfect squares because 9 x 2 = ( 3 x ) 2 and 25 = 5 2 . The polynomial represents a difference of squares and can be rewritten as ( 3 x + 5 ) ( 3 x 5 ) .

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Factor 81 y 2 100.

( 9 y + 10 ) ( 9 y 10 )

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Is there a formula to factor the sum of squares?

No. A sum of squares cannot be factored.

Factoring the sum and difference of cubes

Now, we will look at two new special products: the sum and difference of cubes. Although the sum of squares cannot be factored, the sum of cubes can be factored into a binomial and a trinomial.

a 3 + b 3 = ( a + b ) ( a 2 a b + b 2 )

Similarly, the sum of cubes can be factored into a binomial and a trinomial, but with different signs.

a 3 b 3 = ( a b ) ( a 2 + a b + b 2 )

We can use the acronym SOAP to remember the signs when factoring the sum or difference of cubes. The first letter of each word relates to the signs: S ame O pposite A lways P ositive. For example, consider the following example.

x 3 2 3 = ( x 2 ) ( x 2 + 2 x + 4 )

The sign of the first 2 is the same as the sign between x 3 2 3 . The sign of the 2 x term is opposite the sign between x 3 2 3 . And the sign of the last term, 4, is always positive .

Sum and difference of cubes

We can factor the sum of two cubes as

a 3 + b 3 = ( a + b ) ( a 2 a b + b 2 )

We can factor the difference of two cubes as

a 3 b 3 = ( a b ) ( a 2 + a b + b 2 )

Questions & Answers

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
Ken Reply
proof
AUSTINE
sebd me some questions about anything ill solve for yall
Manifoldee Reply
how to solve x²=2x+8 factorization?
Kristof Reply
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
SO THE ANSWER IS X=-8
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
1KI POWER 1/3 PLEASE SOLUTIONS
Prashant Reply
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
which of these functions is not uniformly continuous on 0,1
Basant Reply
solve this equation by completing the square 3x-4x-7=0
Jamiz Reply
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
Jean Reply
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
Rubben Reply
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
Pab Reply
More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
cool
Rubben
kya
Khushnama
please I need help in maths
Dayo Reply
Okey tell me, what's your problem is?
Navin
Practice Key Terms 2

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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